Toggling between expansions of RS-frames and Urquhart spaces

RS-frames and Urquhart spaces are two approaches to provide dualities for (not necessarily distributive) bounded lattices. The former uses the canonical extension of the original lattice and it is in fact a skeleton of the canonical extension. The later is based directly on the original lattice using topology, but if we drop the topology, the resulting structure (a doset---doubly ordered set) is equivalent to the RS-frame of the canonical extension of the original lattice. In this talk, we will make that correspondence explicit by defining a doset for each RS-frame and a RS-frame for certain dosets. Then each of those dosets is equivalent to its RS-frame (in the sense that by toggling it twice we reach an isomorphic doset) and we identify a condition for a RS-frame to be equivalent to its doset too. The method of toggling between these two frameworks is then expanded to an additional operation (in this case, a negation), based on earlier work on dualities for lattices with negation via Urquhart duality by W. Dzik, E. Orlowska and C. van Alten and similar work (for the same classes of algebras using canonical extensions) by the present speaker. This correspondence enabled us to sort out a problem posed by the former authors in their paper.